Polynomial intersection conjecture for curve graph distance

Let $S$ be a surface with complexity $\xi(S)$, let $k$ be an integer, and let $\alpha$ and $\beta$ be simple closed curves on $S$. Write $d_{\mathcal{C}}$ for curve graph distance and $i(\alpha,\beta)$ for geometric intersection number. Polynomial intersection conjecture. For each $k$, there exists a polynomial $f_k:\mathbb{N}\to\mathbb{N}$ of degree $k-2$ such that

$$d_{\mathcal{C}}(\alpha,\beta)\geq k \Rightarrow i(\alpha,\beta)\geq f_k(\xi(S)).$$

This would strengthen the lower bound used to obtain uniform hyperbolicity of the curve graph. The source does not provide resolution evidence, so the conjecture is recorded as open.